Philippe Mexes opened the scoring for the Giallorossi before the break and Simone Perrotta doubled their advantage soon after the interval.
Pele came on for Inter and had an immediate impact, outfoxing Doni from nearly 30 yards on 59 minutes to set up a tense finish, while Mirko Vucinic and Nicolas Burdisso were sent off.
The victory marks Roma's ninth ever Coppa Italia and caps off a fine season for the Giallorossi.
Roma chief Rosella Sensi beamed: "This is for the president (Franco Sensi). It was an important victory, deserved much as the Scudetto.
"The takeover? We have already commented through a press release."